Statements (25) There is a unique natural number $n$ such that $n ^ { 2 } + 19 n - n ! = 0$. (26) There are infinitely many pairs $( x , y )$ of natural numbers satisfying $$( 1 + x ! ) ( 1 + y ! ) = ( x + y ) ! .$$ (27) For any natural number $n$, consider GCD of $n ^ { 2 } + 1$ and $( n + 1 ) ^ { 2 } + 1$. As $n$ ranges over all natural numbers, the largest possible value of this GCD is 5. (28) If $n$ is the largest natural number for which 20! is divisible by $80 ^ { n }$, then $n \geq 5$.
\textbf{Statements}
(25) There is a unique natural number $n$ such that $n ^ { 2 } + 19 n - n ! = 0$.\\
(26) There are infinitely many pairs $( x , y )$ of natural numbers satisfying
$$( 1 + x ! ) ( 1 + y ! ) = ( x + y ) ! .$$
(27) For any natural number $n$, consider GCD of $n ^ { 2 } + 1$ and $( n + 1 ) ^ { 2 } + 1$. As $n$ ranges over all natural numbers, the largest possible value of this GCD is 5.\\
(28) If $n$ is the largest natural number for which 20! is divisible by $80 ^ { n }$, then $n \geq 5$.